A Novel Switching Table for a Modified Three-Level Inverter-Fed DTC Drive with Torque and Flux Ripple Minimization

Yashar Farajpour 1,* , Mohamad Alzayed 1 , Hicham Chaoui 1 and Sousso Kelouwani 2 1 Intelligent Robotic and Energy Systems (IRES) Research Group, Department of Electronics, Carleton University, Ottawa, ON K1S 5B6, Canada; mohamad.alzayed@carleton.ca (M.A.); hicham.chaoui@carleton.ca (H.C.) 2 Hydrogen Research Institute and the Department of Mechanical Engineering, Université du Québec à Trois-Rivières, Trois-Rivières, QC G8Z 4M3, Canada; sousso.kelouwani@uqtr.ca * Correspondence: yashar.farajpour@carleton.ca


Introduction
The conventional direct torque control (DTC) drive has been investigated and developed as a replacement for field-oriented control (FOC) in high-performance AC motor drives [1]. DTC is well-known for its fast torque response, which requires no coordinate transformation, current control or modulation techniques [2]. Furthermore, the simple architecture and robustness against parameter variations are advantages of this approach [3,4]. The inherent benefits of DTC have made it a popular solution for a wide variety of applications, from brushless DC [5] to permanent magnets [6] and switch reluctance motors [7]. Considering the fact that rotor speed encoders are expensive, can have high maintenance costs and are not suitable for harsh environments, numerous sensorless DTC strategies have been proposed [8][9][10]. Although torque hysteresis controller-based DTC (THC-DTC) is considered to be a fast dynamic response controller, its main drawbacks are high torque ripple and flux droop in the low-speed region [11,12]. Torque ripple has a direct relationship with mechanical loss and tensions on the motor shaft, while flux ripple indicates harmonics and a distorted current waveform.
Researchers have addressed these weaknesses in many works by introducing various strategies; one of these strategies is space vector modulation for classical DTC (DTC-SVM) [13]. However, its reference voltage estimation is a complicated process [14] and advanced control strategies and provides neutral point balance. Second, the vectors are generated by the specific gates that require fewer switching occurrences. This can be achieved by retaining the same switching states. Third, the stator flux plane is divided into 12 controlling sectors, which allows a greater controlling resolution and doubles the possible solutions for vector selection. Fourth, the torque and flux hysteresis controllers are replaced with five and three-level controllers, respectively. This modification means that the drive is capable of discriminating between smaller and larger torque errors and simultaneously adds a constant flux region. Thus, the expectation of this study is decreased torque and flux ripples, as well as harmonic components on the input current, keeping the neutral point of the inverter isolated and providing smooth switching. Furthermore, some important notes are provided that will help in the tuning of a DTC drive.
This paper is organized as follows: we introduce the mathematical model of an induction motor in Section 2. The structure of the proposed model is presented in Section 3 along with the structure of a three-level inverter and its optimized switching table. Afterwards, a series of simulations is presented to illustrate the effectiveness of the proposed model in terms of the different factors of torque ripple, flux ripple, flux trajectory and THD% in Section 4. Ultimately, all the accomplishments and the results are summarized and analyzed in Section 5.

Induction Motor Equations
In order to estimate the torque and stator flux of an induction motor (IM), the first step is to transform the sampled voltage and currents from the abc to the αβ frame using Clarke transformation.
where v abc and v αβ are the input three-phase voltages in abc and αβ frames, respectively. Similarly, i αβ is obtained from i abc measurements. It should be noted that, since v a , v b and v c are all 120 • apart, the third row of the Clarke transformation is zero (as long as the load is balanced). Thus, an isolated neutral The stator voltage in an asynchronous motor can be expressed by a voltage drop in the stator winding and the electromotive force generated by the flux, as expressed in Equation (2). Then, in order to estimate the amplitude and phase of the flux, first, the αβ components of the stator flux need to be calculated with Equation (3), which is derived by integrating Equation (2).
where R s , ϕ αs , ϕ βs and i s are the stator resistance, the flux components and the stator current, respectively. Equation (4) estimates the stator flux magnitude and its angle, and Equation (5) calculates the electromagnetic torque.
where T e is the electromagnetic torque and p represents the pole pairs of the stator winding. The required variables on the motor side are the electromagnetic torque, the stator flux magnitude and its angle, which are all obtained from Equations (4) and (5). Figure 1 depicts a general schematic of a DTC drive. The sampled voltage and current vectors are transformed to the αβ frame using Equation (1), and the main DTC block estimates the torque and stator flux using Equations (4) and (5). A speed controller compares the current rotor speed with the desired value and generates two reference values for torque and flux. Then, two hysteresis controllers produce the required commands for the switching table to satisfy the demanded rotor speed by contrasting the actual and reference values of torque and flux. Thus, the decoupled control of torque and flux can be achieved in DTC. The simplicity and independency of motor parameters are decisive factors in the popularity of DTC. Likewise, our proposed method does not include any additional motor parameters in the control loop, and it does not affect the system with an extra computational burden. However, it is updated with a 12-sector stator flux plane, higher levels of torque and flux hysteresis controllers, a three-level inverter and an optimized switching table that coordinates and exploits all these advantages. Figure 2 illustrates the structure of an NPC 3L inverter. Figure 3 shows how the 19 unique voltage vectors of the 3L inverter are distributed on the 12 equally divided sectors of the stator flux plane. The proposed method allocates at least one voltage vector to each section to increase control resolution. In contrast to the conventional method, which needs 60 • to change the flux hysteresis command, the proposed method selects appropriate vectors every 30 • . Besides, the 3L inverter provides us with voltage vectors of different sizes (V dc ,    The process of generating reference torque and flux signals is similar to the conventional DTC, as demonstrated in Figure 4. A proportional-integral controller (PIC) is responsible for translating the rotor speed error into the required torque to satisfy the speed demand. Several other controllers can be employed for this task; however, for the sake of simplicity and in order to provide a fair comparison between different strategies, the PIC is retained. The reference stator flux is kept constant and equal to the nominal flux within the operational speed range (0 to nominal speed). However, at very high speed ranges, flux weakening is required to keep the motor within the constant power region [40]. Thus, when the rotor speed exceeds nominal speed, a flux weakening algorithm reduces the reference flux according to Equation (6).

Proposed DTC Model
where ϕ * s and ϕ sn are the reference and nominal stator fluxes, while ω r and ω n are the actual and nominal rotational speeds, respectively. Figure 5 exhibits five-level torque and three-level flux hysteresis controllers that are replaced with the conventional three-level and two-level torque and flux controllers, respectively. The proposed technique differentiates between small and large torque errors so that it triggers higher voltage levels to compensate larger torque errors and fires smaller vectors for smaller errors. Consequently, the output of the proposed five-level torque hysteresis controller can be (TorqueError = T * e − T e ): • Torque error is large and positive → H T = 2 • Torque error is small and positive → H T = 1 • Torque error is acceptable → H T = 0 • Torque error is small and negative → H T = −1 • Torque error is large and negative → H T = −2 According to Figure 3, there are at least two vectors that are perpendicular to each section, which enables us to add the third level for the flux hysteresis controller. Thus, its outputs are defined as It must be taken into consideration that it is not possible to reduce the torque or flux ripples only by tightening the torque or flux hysteresis bands. A very wide band will result in more ripples and even produce insufficient amounts of torque in some instances. Furthermore, tightening the torque or flux hysteresis bands can result in a significant increase in the switching frequency. Mitigating the ripples mostly depends on the sampling time of the control system and the magnitude of the injected voltage vectors. Trying to compensate a small torque error with a large voltage vector will result in an overshoot from the desired limits in a given sampling time; thus, attempting to alter the hysteresis bands is futile. In other words, if the hysteresis bands are set to be too narrow, not only will the ripples not decrease, but also the switching frequency (i.e., switching losses) will increase for no gain. Thus, adjusting the optimized hysteresis bands is of foremost importance, although this fact is unfortunately neglected by many researchers.
Eventually, the conventional switching table will no longer be applicable and a novel switching table will be needed to utilize the expressed model. Considering the 19 unique voltage vectors of the three-level inverter, the 12 flux sectors, the five levels for torque commands and three levels for flux commands, the proposed optimized switching table for a three-level inverter-equipped DTC drive is presented in Table 1. Evidently, the preference is for selecting smaller vectors (V 13 → V 18 ), while bigger vectors (V 1 → V 12 ) are used for the compensation of considerable deviations from the tolerable torque error. It is believed that, in most of the steady-state conditions, half of the DC link voltage is enough to satisfy the user demand. At the same time, the full magnitude vectors are generally needed only during the transient states to provide higher electromagnetic torque.
It is noteworthy that, according to Figure 3, there are two possible switching patterns for generating vectors V 13 to V 18 . For the purpose of smooth switching and preventing voltage jumps across the inverter gates, and considering the fact that the proposed model generally injects zero or half of the DC voltage, the 0 and 1 states of the switches are chosen in the switching  14 [221] means a half DC link voltage increase in the first and the third legs, and a full voltage jump in the second leg. The proposed optimized switching table follows this rule in order to prevent voltage jumps and simultaneously to decrease the switching frequency. Besides, the switching pattern is organized in such a way that it maintains the neutral point balance and ensures zero current in the neutral point. According to the above-mentioned considerations, V 0 and V 13 to V 18 are triggered using these states: where, V 0 [1 1 1] acts as an intermediate that prevents voltage jumps between states 0 and 2.

Voltage Vector Synthesis Strategy
In order to provide a fair comparison with another DTC technique that utilizes a 3L inverter, a vector synthesis method was chosen [38]. As discussed in the first section, this is one of the best-known strategies, especially for 3L inverter-fed DTC drives. This method utilizes all the voltage vectors of a 3L inverter and produces the desired synthesized vectors at different angles. This method secures smooth switching, maintains neutral point balance and has been proposed for ripple reduction purposes. Figure 6 illustrates the synthesized voltage vectors that are produced with the vector synthesis sequence presented in Table 2. The design and implementation details of the vector synthesis strategy are laid out in [38].  Figure 6. Synthesis vector diagram [38]. Table 2. Switching sequences used to generate synthesized voltage vectors [38]. As stated in Table 2, in every transition, only one state changes. In order to prevent voltage jumps, every sequence begins with [111]. To synthesize each vector, eight voltage vectors are triggered with equal duty ratios. Thus, the eight sampling times last until a vector is synthesized. Unlike the conventional DTC methods, the switching frequency is constant for this technique and is determined by the duty ratio of each vector. This model is equipped with two three-level hysteresis controllers to define both torque and flux commands. The switching table is provided in Table 3, where k is the number of the estimated flux sector. It is worthy of mention that this model requires a smaller sampling time for the switching circuit and an up to three times higher switching frequency [38]. Table 3. Switching table of vector synthesis DTC [38].

Simulation Results and Discussion
A series of comprehensive simulations were carried out in the MATLAB ® platform to show the effectiveness of the proposed strategy on a 10 horsepower (HP) induction motor. In this step, the conventional 2L inverter-fed DTC, the vector synthesis DTC fed by a 3L inverter and our proposed method were studied side-by-side under the same conditions. However, due to the fact that the synthesis vector method required fast vector injection, the sampling time for this method was set to 10 microseconds, while the conventional DTC and our proposed model were simulated for 30 microseconds. Figure 7 shows a high-level schematic of the simulation setup. It also highlights the differences in hysteresis controllers, switching tables, vector selection algorithms and voltage source inverters. All three models were equipped with the same very well-tuned PI speed controller. The hysteresis bands for all models were adjusted meticulously in a way that provoked minimum torque and flux ripples and guaranteed stable operation. The performances of the examined models were analyzed in very low (20 rpm), normal (1000 rpm) and high (2000 rpm) rotor speed demands with an acceleration ramp of 2000 (rpm/s) under no-load and load-torque disturbances. They were studied in terms of the torque, flux, voltage, current and THD%.     15 N·m was mounted on the shaft, which was then decoupled at t = 2 s. Low speed demands such as 20 rpm create very challenging situations for DTC drives. However, choosing the right reference flux and well-tuned hysteresis controllers help to meet very low speed demands. As is shown in Figure 8a-c, in a fraction of a second, the rotor reached the reference speed with no overshoot. When the load was applied (t = 1 s) and removed (t = 2 s), there were small deviations from the reference of 3 rpm, which were compensated in less than half a second. Since the same speed controller operated all the models, their speed responses were very similar except that of the proposed method, which was less noisy.  Figure 8j-l. Besides this, it is an indication of lower harmonic distortion (THD%), which will be discussed later. Figure 8m-o completely differentiates these methods by exhibiting their voltage injection preferences. As stated earlier, the switching table of the proposed method is able to differentiate between large and small torque errors, and it tries to compensate smaller torque errors with smaller voltage vectors, while it switches the larger voltage vectors when it is necessary. Likewise, Figure 8o shows that higher voltage amplitudes were only needed for a fraction of a second in starting condition, and the proposed drive was capable of providing precise torque, flux and speed responses with lower voltage levels even under the condition of full load-torque disturbance.

Normal-Speed Performance; 1000 rpm
In this step, the drives started with a speed demand of 1000 rpm. A slight overshoot occurred when they reached the reference at t = 0.5 s. A 15 N·m load was applied to the shaft, and then removed at t = 2 s and t = 3 s, respectively. Figure 9a-c shows the comparable speed responses for all cases with two 3 rpm deviations each for half a second when the load disturbances occurred. Figure 9d-f shows the similar torque responses to those for the low-speed performance for each strategy. The torque ripples were 10, 6, and 3 N·m for conventional DTC, vector synthesis and the proposed method, respectively. During the normal operational range, Figure 9g

High-Speed Performance; 2000 rpm
Finally, the drives were tested at the high speed of 2000 rpm under no-load (up to t = 2s) and load-torque conditions (t = 2 → 3 s). Although the speed responses, as shown in Figure 10a-c, were similar, the torque responses showed some important details. As mentioned previously, the switching frequency of the vector synthesis method is constant because it injects a slightly different vector at every sampling time to synthesize the desired vector. Thus, the torque response of this method was similar to that demonstrated earlier. The torque ripple of the conventional DTC dropped from 10 to 9 N·m when it exceeded the nominal synchronous speed of 1500 rpm (for a four-pole 50 Hz motor). This method operates based on a variable switching frequency, which mostly depends on the hysteresis bands and rotor speed. Thus, when the rotor speed increases, the switching frequency increases as well. Consequently, the torque ripple decreases to some extent. On the other hand, the torque ripple of the proposed method increased from 3 N·m to 4 N·m when it passed the operational speed range. Figure 10o explains the reason for this by revealing a change in the behavior of the switching table, as it triggered higher voltage levels more frequently in the high-speed region (after t = 0.7 s) when the motor entered the constant power region. However, the torque ripple for the proposed method was still lower than the others. Figure 10g-i illustrates the stator fluxes and the effect of the flux weakening algorithm as the rotor speed surpassed the operational range. The flux ripples of the conventional DTC and vector synthesis method experienced a small reduction, to 0.04 and 0.015 Wb, respectively, while the proposed method still maintained this value below 0.01 Wb. Ultimately, Figure 11 illustrates the effectiveness of the proposed technique in maintaining the neutral point balance even during the severe condition of high-speed operation and load-torque disturbance. Because the designated vectors in the switching table were chosen meticulously, voltage unbalance did not occur and the current in ground connection was almost zero (in the 10 −16 range). The same figure also demonstrates the voltage change over one of the inverter gates that did not exceed half of the DC link voltage under the same aforementioned conditions. Figure 12 depicts the flux trajectories in αβ axes, which is considered to be an excellent representation of the harmonic distortions in a system. The more circular a flux trajectory, the lower the THD% it produces in the system. Figure 13 shows the THD% of the investigated methods at various rotor speeds (from very low to high) and under no-load and load-torque conditions. It was expected that when a load was coupled, as the input current increased, the THD% would decrease, because the share of the fundamentals increased. Since a 3L inverter produces a more sinusoidal current, the harmonics decrease. However, the proposed model contaminates the network with much lower THD% than the vector synthesis method in most operating conditions.

Conventional DTC
Vector Synthesis Proposed Method

Conclusions
In this study, we propose a novel technique for a 3L inverter-fed DTC drive. Our method provides an optimized switching table capable of discriminating between large and small torque errors and introducing a third dimension for stator flux control. It requires the use of new hysteresis controllers that allow a greater resolution. Thus, flux and torque commands are issued in three and five levels, respectively. The proposed switching table is organized in such a way to prevent voltage jumps by utilizing the inverter gates with only half of the DC link voltage. Besides this, this model ensures neutral point isolation by providing a voltage balance for each phase. Even for the generation of some vectors for which there are more than one switching pattern, 0 and 1 states are preferred to ensure minimum state changes, resulting in a lower switching frequency. The proposed technique was compared with a conventional model and the vector synthesis DTC model, which has been investigated in great depth for its ripple reduction and smooth switching characteristics. Owing to the fact that all three models used similar well-tuned PI speed controllers, the speed responses were comparable and very accurate. All models demonstrated consistent performances at low, medium, and high speeds (from 20 to 2000 rpm) and under load-torque disturbances. The proposed model showed its efficiency by minimizing the torque ripple by up to 70% and 50% compared with the conventional and vector synthesis DTC models. Its flux ripple was also 78% smaller than the conventional DTC and 50% lower than the vector synthesis method. It also improved THD% and reduced harmonics. However, thethe proposed model runs with a three-times longer sampling delay than the vector synthesis method, which samples every 10 microseconds. This model will help to reduce maintenance costs and mechanical loss by reducing the torque ripple and the stresses on the motor shaft, and it leaves fewer detectable harmonic traces in the network.
Since a three-level inverter is employed, the switching losses are higher compared to the two-level inverter case because the number of IGBTs is doubled. It is noteworthy that, considering similar hysteresis bands, the switching losses for the proposed model will be less than the two-level inverter-fed model as it manages the motor with lower voltage levels, meaning that the switching frequency could also be lower. However, due to the fact that the aim of the method is to reduce the ripples, the hysteresis bands must be chosen accurately. In some cases, there might be a need for a compromise between favoring lower ripple or lower switching losses. However, the proposed model has an advantage over the other three-level inverter-equipped techniques as it prefers lower voltage levels. It is worthy of mention that this advantage is valid for the operational speed range and under the nominal load condition. If the demanded speed exceeds the nominal speed, the proposed model behaves similarly to the other three-level inverter DTC models, as shown in Figure 10o. For future studies, we suggest investigating the experimental results and analyzing the motor and inverter losses, considering the fact that this model operates with lower voltage and current amplitudes for a given motor. Due to the fact that this model does not require additional equipment, the only concern is the precise tuning of the hysteresis bands and the dead-time of the inverter when considering the sampling time of the control loop.